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Rational numbers were introduced, because they allow to solve equations of the type q x = p , x = p q , q โ 0 , p , q โ โค We can imagine x to be an ordered pair x = (p, q). Such numbers are also called fractions or quotients. The representation of a rational number is not unique! 3 6
Fractions can be reduced, or the numerator and the denominator can be multiplied by the same factor, without changing the value of the fraction. The representation is unique only, if numerator and denomi- nator are relatively prime. p 1 = p (^) โ therefore the rational numbers include the integers.
Rational numbers Rational numbers
Ending decimal fractions : Repeating decimal fractions : 3 5 = 0.6 , 7 4 = 1. 1 3 = 0.333 ๎ = 0. 3 , 19 9 = 2. 111 ๎ = 2. 1 The period is marked by placing a bar over the repeating decimal digits (or possibly underlining them) :
- 7 = 3 ๎ 7 10 ๎ 7 10 2 ๎^ 7 10 3 ๎^ 7 10 4 ๎^ ๎ Statement: Each rational number can be represented by an ending or repeating decimal fraction. Conversely, each ending or repeating decimal fraction can be represented as fraction p /q. 0.05 31 = 5 10 2 ๎ 31 10 4 ๎ 31 10 6 ๎ 31 10 8 ๎ ๎
Rational numbers Rational numbers
SquareSquare root ofroot of twotwo
x 2 = 2 , x = ยฑ (^) ๎ 2 What is โ2? ๎^2 โ^ 1.41,^ 1. 2 = 1.9881 ; (^) ๎ 2 โ 1.414 , 1. 2 = 1. x = 1,4142135623730950488016887242096980785696718753_..._ 699 We may use MAPLE, to calculate the decimal expansion of โ2 to 415 digits: x 2 = 1,9999999999999999999999999999999999999999999999_..._ 010 This number satisfies the equation xยฒ = 2 to high precision, but not exactly. By finite decimal expansion, we will never get a number, the square of which is exactly 2.
The length of the diagonal of a square with sides of length 1 is โ2, as indicated in the figure. It can not be written as fraction of two integers. That is, โ2 is not a rational number (geometrically shown al- ready around 500 BC, with numbers about 200 years later by Euclid). โฃ AC โฃ 2 = โฃ AB โฃ 2 ๎ โฃ BC โฃ 2 = 1 2 ๎ 1 2
= 2 โ โฃ AC โฃ = ๎ 2
๎^2
A B
C
1 1 x Fig 5: The number โ2 as diagonal and on the number line
Square Square root ofroot of twotwo
Irrational numbersIrrational numbers
- http://i032.radikal.ru/0804/82/af2caf626175.jpg Irrational numbers are infinite, not repeating decimal fractions. Most of the outputs of the root function, of the logarithmic functions or trigonometric functions, as well as the numbers and e are irratio- nal numbers. ๎ Archimedes' constant (^) ๎ = 3.141592654 ๎ Euler's number (^) e = 2,718281828459 ๎
- http://images.zeit.de/bilder/2007/24/wissen/wissenschaft/euler/euler-artikel.jpg 1) 2)
Real numbersReal numbers
The sets of rational and irrational numbers form together the set of real numbers_._ The set of real numbers can be seen as the set of all points on the number line. The points corresponding to real numbers cover the line completely. โ addition โ multiplication โ subtraction (existence of additive inverse) โ division (existence of multiplicative inverse) โ order relation โ Operations and relations on the set โof real numbers :
RealReal numbers:numbers: intervalsintervals
a b
- [ a , โ ) = { x | a ๎ x ๎ โ}
- ๎ a , โ๎ = { x | a ๎ x ๎ โ}
- ๎โโ , b ๎ = { x | โโ ๎ x ๎ b } โ = ๎โโ , โ๎
